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  • 1.
    Elmukashfi, Elsiddig
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    A multiscale continuum modeling of strain-induced cavitation damageand crystallization in rubber-like materials2015Report (Other academic)
    Abstract [en]

    A multiscale continuum model for strain-induced cavitation damage and crystallization for rubber-like materials is proposed. The constitutive behavior is determined by homogenization over different length scales, namely, the nano-scale, micro-scale and macro-scale. The microstructure of a filled rubber-like material is seen as interaction between clusters of the filler particles and long-chain molecules that form two networks, between cross-links and between the filler aggregates. The network between cross-links in the nano-scale is modeled using the full network approach of semi-crystalline chains. A phenomenological law is proposed to describe the crystallite nucleation law. The network between the filler particles is described by statistical mechanics in the nano- and/or micro-scale where the polymer chains sliding on and/or debonding from filler aggregate surface is incorporated. The debonding process is regarded as the main mechanism of the nucleation of nano-cavities which introduces non-affine deformation to the network between cross-links. Hence, The nanoscopic initially non-cavitated network between cross-links is homogenized over the micro-scale assuming a spherical representative volume element using the kinematics proposed by Hang-Sheng and Abeyaratne (Hang-Sheng and Abeyaratne in Journal of the Mechanics and Physics of Solids 40 (3), 571592,1992). The constitutive model is presented in form of an averaged strain energy function. The predictive capabilities of the model are then tested via comparisons with experimental data from literature.

  • 2.
    Elmukashfi, Elsiddig
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Numerical analysis of dynamic crack propagation in biaxially strained rubber sheets2013Report (Other academic)
  • 3.
    Elmukashfi, Elsiddig
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Numerical analysis of dynamic crack propagation in biaxially strained rubber sheets2014In: Engineering Fracture Mechanics, ISSN 0013-7944, E-ISSN 1873-7315, Vol. 124, p. 1-17Article in journal (Refereed)
    Abstract [en]

    This paper proposes a computational framework for dynamic crack propagation in rubber in which a nonlinear finite element analysis using cohesive zone modeling approach is used. A suddenly initiated crack at the center of biaxially stretched sheet problem is studied under plane stress conditions. A transient dynamic analysis using implicit time integration scheme is performed. In the constitutive modeling, the continuum is characterized by finite-viscoelasticity theory and coupled with the fracture processes using a cohesive zone model. This computational framework was introduced previously by the present authors (Elmukashfi and Kroon, 2012). In the current work, the use of a rate-dependent cohesive model is examined in addition to investigation of generalized biaxial loading cases. A Kelvin-Voigt element is used to describe the rate-dependent cohesive model wherein the spring is described by a bilinear law and dashpot with a constant viscosity is adopted. An explicit integration is used to incorporate the rate-dependent cohesive model in the finite element environment. A parametric study over the cohesive viscosity is performed and the steady crack propagation velocity is evaluated and compared with experimental data. It appears that the viscosity varies with the crack speed. Further, the total work of fracture is estimated using rate-independent cohesive law such that the strength of the cohesive zone is assumed to be constant and the separation work per unit area is determined form the experimental data. The results show that fracture-related processes, i.e. creation of new surfaces, cavitation and crystallization; contribute to the total work of fracture in a contradictory manner.

  • 4.
    Elmukashfi, Elsiddig
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Numerical analysis of dynamic crack propagation in rubber2012In: International Journal of Fracture, ISSN 0376-9429, E-ISSN 1573-2673, Vol. 177, no 2, p. 163-178Article in journal (Refereed)
    Abstract [en]

    In the present paper, dynamic crack propagation in rubber is analyzed numerically using the finite element method. The problem of a suddenly initiated crack at the center of stretched sheet is studied under plane stress conditions. A nonlinear finite element analysis using implicit time integration scheme is used. The bulk material behavior is described by finite-viscoelasticity theory and the fracture separation process is characterized using a cohesive zone model with a bilinear traction-separation law. Hence, the numerical model is able to model and predict the different contributions to the fracture toughness, i.e. the surface energy, viscoelastic dissipation, and inertia effects. The separation work per unit area and the strength of the cohesive zone have been parameterized, and their influence on the separation process has been investigated. A steadily propagating crack is obtained and the corresponding crack tip position and velocity history as well as the steady crack propagation velocity are evaluated and compared with experimental data. A minimum threshold stretch of 3.0 is required for crack propagation. The numerical model is able to predict the dynamic crack growth. It appears that the strength and the surface energy vary with the crack speed. Finally, the maximum principal stretch and stress distribution around steadily propagation crack tip suggest that crystallization and cavity formation may take place.

  • 5.
    Elmukashfi, Elsiddig
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    Numerical modeling and analysis of dynamic crack propagation in rubber2013In: 13th International Conference on Fracture 2013, ICF 2013: Volume 5, 2013, Chinese Society of Theoretical and Applied Mechanics , 2013, p. 3598-3607Conference paper (Refereed)
    Abstract [en]

    Dynamic crack propagation in rubber is modeled and analyzed numerically using the finite element method. The problem of a suddenly initiated crack at the center of stretched sheet is studied under plane stress conditions. A nonlinear finite element analysis using implicit time integration scheme is used. The bulk material behavior is described by finite-viscoelasticity theory and the fracture separation process is characterized using a cohesive zone model with a bilinear traction-separation law. Hence, the numerical model is able to model and predict the different contributions to the fracture toughness, i.e. the surface energy, viscoelastic dissipation, and inertia effects. The separation work per unit area and the cohesive strength has been parameterized, and their influence on the separation process has been investigated. A steadily propagating crack is obtained and the corresponding crack tip position and velocity history as well as the steady crack propagation velocity are evaluated and compared with the experimental data. A minimum threshold stretch of 3.0 is required for crack propagation. The numerical model is able to predict the dynamic crack growth such that the strength and the surface energy vary with the crack speed.

  • 6.
    Eriksson, Thomas
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Holzapfel, Gerhard A.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Influence of Medial Collagen Organization and Axial In Situ Stretch on Saccular Cerebral Aneurysm Growth2009In: Journal of Biomechanical Engineering, ISSN 0148-0731, E-ISSN 1528-8951, Vol. 131, no 10Article in journal (Refereed)
    Abstract [en]

    A model for saccular cerebral aneurysm growth, proposed by Kroon and Holzapfel (2007, "A Model for Saccular Cerebral Aneurysm Growth in a Human Middle Cerebral Artery," J. Theor. Biol., 247, pp. 775-787; 2008, "Modeling of Saccular Aneurysm Growth in a Human Middle Cerebral Artery," ASME J. Biomech. Eng., 130, p. 051012), is further investigated. A human middle cerebral artery is modeled as a two-layer cylinder where the layers correspond to the media and the adventitia. The immediate loss of media in the location of the aneurysm is taken to be responsible for the initiation of the aneurysm growth. The aneurysm is regarded as a development of the adventitia, which is composed of several distinct layers of collagen fibers perfectly aligned in specified directions. The collagen fibers are the only load-bearing constituent in the aneurysm wall; their production and degradation depend on the stretch of the wall and are responsible for the aneurysm growth. The anisotropy of the surrounding media was modeled using the strain-energy function proposed by Holzapfel et al. (2000, "A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models," J. Elast., 61, pp. 1-48), which is valid for an elastic material with two families of fibers. It was shown that the inclusion of fibers in the media reduced the maximum principal Cauchy stress and the maximum shear stress in the aneurysm wall. The thickness increase in the aneurysm wall due to material growth was also decreased. Varying the fiber angle in the media from a circumferential direction to a deviation of 10 deg from the circumferential direction did, however, only show a little effect. Altering the axial in situ stretch of the artery had a much larger effect in terms of the steady-state shape of the aneurysm and the resulting stresses in the aneurysm wall. The peak values of the maximum principal stress and the thickness increase both became significantly higher for larger axial stretches. [DOI: 10.1115/1.3200911]

  • 7.
    Faleskog, Jonas
    et al.
    KTH, Superseded Departments, Solid Mechanics.
    Kroon, Martin
    KTH, Superseded Departments, Solid Mechanics.
    Öberg, Hans
    KTH, Superseded Departments, Solid Mechanics.
    A probabilistic model for cleavage fracture with a length scale - parameter estimation and predictions of stationary crack experiments2004In: Engineering Fracture Mechanics, ISSN 0013-7944, E-ISSN 1873-7315, Vol. 71, no 1, p. 57-79Article in journal (Refereed)
    Abstract [en]

    This study presents a large experimental investigation in the transition temperature region on a modified A508 steel. Tests were carried out on single-edge-notch-bend specimens with three different crack depth over specimen width ratios to capture the strong constraint effect on fracture toughness. Three test temperatures were considered, covering a range of 85 degreesC. All specimens failed by cleavage fracture prior to ductile tearing. A recently proposed probabilistic model for the cumulative failure by cleavage was applied to the comprehensive sets of experimental data. This modified weakest link model incorporates a length scale, which together with a threshold stress reduce the scatter in predicted toughness distributions as well as introduces a fracture toughness threshold value. Model parameters were estimated by a robust procedure, which is crucial in applications of probabilistic models to real structures. The conformity between predicted and experimental toughness distributions, respectively, were notable at all the test temperatures.

  • 8.
    Fallqvist, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Fielden, Matthew
    KTH, School of Engineering Sciences (SCI), Applied Physics.
    Pettersson, Torbjörn
    KTH, School of Chemical Science and Engineering (CHE), Fibre and Polymer Technology, Fibre Technology.
    Nordgren, Niklas
    KTH, School of Chemical Science and Engineering (CHE), Chemistry, Surface and Corrosion Science.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    Gad, Annica
    Karolinska Institutet, MTC.
    Experimental and computational assessment of F-actin influence in regulating cellular stiffness and relaxation behaviour of fibroblasts2016In: Journal of The Mechanical Behavior of Biomedical Materials, ISSN 1751-6161, E-ISSN 1878-0180, Vol. 59, p. 168-184Article in journal (Refereed)
    Abstract [en]

    In biomechanics, a complete understanding of the structures and mechanisms that regulate cellular stiffness at a molecular level remain elusive. In this paper, we have elucidated the role of filamentous actin (F-actin) in regulating elastic and viscous properties of the cytoplasm and the nucleus. Specifically, we performed colloidal-probe atomic force microscopy (AFM) on BjhTERT fibroblast cells incubated with Latrunculin B (LatB), which results in depolymerisation of F-actin, or DMSO control. We found that the treatment with LatB not only reduced cellular stiffness, but also greatly increased the relaxation rate for the cytoplasm in the peripheral region and in the vicinity of the nucleus. We thus conclude that F-actin is a major determinant in not only providing elastic stiffness to the cell, but also in regulating its viscous behaviour. To further investigate the interdependence of different cytoskeletal networks and cell shape, we provided a computational model in a finite element framework. The computational model is based on a split strain energy function of separate cellular constituents, here assumed to be cytoskeletal components, for which a composite strain energy function was defined. We found a significant influence of cell geometry on the predicted mechanical response. Importantly, the relaxation behaviour of the cell can be characterised by a material model with two time constants that have previously been found to predict mechanical behaviour of actin and intermediate filament networks. By merely tuning two effective stiffness parameters, the model predicts experimental results in cells with a partly depolymerised actin cytoskeleton as well as in untreated control. This indicates that actin and intermediate filament networks are instrumental in providing elastic stiffness in response to applied forces, as well as governing the relaxation behaviour over shorter and longer time-scales, respectively.

  • 9.
    Fallqvist, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    A chemo-mechanical constitutive model for transiently cross-linked actin networks and a theoretical assessment of their viscoelastic behaviour2013In: Biomechanics and Modeling in Mechanobiology, ISSN 1617-7959, E-ISSN 1617-7940, Vol. 12, no 2, p. 373-382Article in journal (Refereed)
    Abstract [en]

    Biological materials can undergo large deformations and also show viscoelastic behaviour. One such material is the network of actin filaments found in biological cells, giving the cell much of its mechanical stiffness. A theory for predicting the relaxation behaviour of actin networks cross-linked with the cross-linker alpha-actinin is proposed. The constitutive model is based on a continuum approach involving a neo-Hookean material model, modified in terms of concentration of chemically activated cross-links. The chemical model builds on work done by Spiros (Doctoral thesis, University of British Columbia, Vancouver, Canada, 1998) and has been modified to respond to mechanical stress experienced by the network. The deformation is split into a viscous and elastic part, and a thermodynamically motivated rate equation is assigned for the evolution of viscous deformation. The model predictions were evaluated for stress relaxation tests at different levels of strain and found to be in good agreement with experimental results for actin networks cross-linked with alpha-actinin.

  • 10.
    Fallqvist, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    Kulachenko, Artem
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    Cross-link debonding in actin networks: influence on mechanical properties2015In: International Journal of Experimental and Computational Biomechanics, ISSN 1755-8743, Vol. 3, no 1, p. 16-26, article id b778558v5j17h4n8Article in journal (Refereed)
    Abstract [en]

    The actin cytoskeleton is essential for the continued function and survival of the cell. A peculiar mechanical characteristic of actin networks is their remodelling ability, providing them with a time-dependent response to mechanical forces. In cross-linked actin networks, this behaviour is typically tuned by the binding affinity of the cross-link. We propose that the debonding of a cross-link between filaments can be modelled using a stochastic approach, in which the activation energy for a bond is modified by a term to account for mechanical strain energy. By use of a finite element model, we perform numerical analyses in which we first compare the model behaviour to experimental results. The computed and experimental results are in good agreement for short time scales, but over longer time scales the stress is overestimated. However, it does provide a possible explanation for experimentally observed strain-rate dependence as well as strain-softening at longer time scales.

  • 11.
    Fallqvist, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Kulachenko, Artem
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Modelling of cross-linked actin networks - Influence of geometrical parameters and cross-link compliance2014In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 350, p. 57-69Article in journal (Refereed)
    Abstract [en]

    A major structural component of the cell is the actin cytoskeleton, in which actin subunits are polymerised into actin filaments. These networks can be cross-linked by various types of ABPs (Actin Binding Proteins), such as Filamin A. In this paper, the passive response of cross-linked actin filament networks is evaluated, by use of a numerical and continuum network model. For the numerical model, the influence of filament length, statistical dispersion, cross-link compliance (including that representative of Filamin A) and boundary conditions on the mechanical response is evaluated and compared to experimental results. It is found that the introduction of statistical dispersion of filament lengths has a significant influence on the computed results, reducing the network stiffness by several orders of magnitude. Actin networks have previously been shown to have a characteristic transition from an initial bending-dominated to a stretching-dominated regime at larger strains, and the cross-link compliance is shown to shift this transition. The continuum network model, a modified eight-chain polymer model, is evaluated and shown to predict experimental results reasonably well, although a single set of parameters cannot be found to predict the characteristic dependence of filament length for different types of cross-links. Given the vast diversity of cross-linking proteins, the dependence of mechanical response on cross-link compliance signifies the importance of incorporating it properly in models to understand the roles of different types of actin networks and their respective tasks in the cell.

  • 12.
    Fallqvist, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    Martin, Kroon
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    Constitutive modelling of composite biopolymer networks2016In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 395, p. 51-61Article in journal (Refereed)
    Abstract [en]

    The mechanical behaviour of biopolymer networks is to a large extent determined at a microstructural level where the characteristics of individual filaments and the interactions between them determine the response at a macroscopic level. Phenomena such as viscoelasticity and strain-hardening followed by strain-softening are observed experimentally in these networks, often due to microstructural changes (such as filament sliding, rupture and cross-link debonding). Further, composite structures can also be formed with vastly different mechanical properties as compared to the indivudal networks. In this present paper, we present a constitutive model presented in a continuum framework aimed at capturing these effects. Special care is taken to formulate thermodynamically consistent evolution laws for dissipative effects. This model, incorporating possible anisotropic network properties, is based on a strain energy function, split into an isochoric and a volumetric part. Generalisation to three dimensions is performed by numerical integration over the unit sphere. Model predictions indicate that the constitutive model is well able to predict the elastic and viscoelastic response of biological networks, and to an extent also composite structures.

  • 13.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    A constitutive framework for modelling thin incompressible viscoelastic materials under plane stress in the finite strain regime2011In: Mechanics of time-dependant materials, ISSN 1385-2000, E-ISSN 1573-2738, Vol. 15, no 4, p. 389-406Article in journal (Refereed)
    Abstract [en]

    Rubbers and soft biological tissues may undergo large deformations and are also viscoelastic. The formulation of constitutive models for these materials poses special challenges. In several applications, especially in biomechanics, these materials are also relatively thin, implying that in-plane stresses dominate and that plane stress may therefore be assumed. In the present paper, a constitutive model for viscoelastic materials in the finite strain regime and under the assumption of plane stress is proposed. It is assumed that the relaxation behaviour in the direction of plane stress can be treated separately, which makes it possible to formulate evolution laws for the plastic strains on explicit form at the same time as incompressibility is fulfilled. Experimental results from biomechanics (dynamic inflation of dog aorta) and rubber mechanics (biaxial stretching of rubber sheets) were used to assess the proposed model. The assessment clearly indicates that the model is fully able to predict the experimental outcome for these types of material.

  • 14.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    A constitutive model for smooth muscle including active tone and passive viscoelastic behaviour2010In: Mathematical Medicine and Biology, ISSN 1477-8599, E-ISSN 1477-8602, Vol. 27, no 2, p. 129-155Article in journal (Refereed)
    Abstract [en]

    A new constitutive model for the biomechanical behaviour of smooth muscle tissue is proposed. The active muscle contraction is accomplished by the relative sliding between actin and myosin filaments, comprising contractile units in the smooth muscle cells. The model includes a chemical part, governing the cross-bridge (myosin head) cycling, that is responsible for the filament sliding. The number of activated cross-bridges govern the contractile force generated and also the contraction speed. A strain-energy function is used to describe the mechanical behaviour of the smooth muscle tissue. Besides the active contractile apparatus, the mechanical model also incorporates a passive viscoelastic part. The constitutive model was calibrated with respect to experiments on smooth muscle tissue from swine carotid artery and guinea pig taenia coli, in terms of isometric and isotonic tensile test results. The model was fully able to reproduce the experimental results.

  • 15.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    A constitutive model for strain-crystallising Rubber-like materials2010In: Mechanics of materials (Print), ISSN 0167-6636, E-ISSN 1872-7743, Vol. 42, no 9, p. 873-885Article in journal (Refereed)
    Abstract [en]

    In the present paper, a constitutive model for strain-crystallising rubber is proposed. The constitutive behaviour is formulated in terms of a strain energy function, where the full network approach is adopted. The Arrhenius equation provides the basis for the crystallite nucleation law. The full network approach allows for the development of an anisotropic crystal structure. The model was applied to experimental results from uniaxial tensile tests. Strain-crystallisation causes a hysteresis in the stress-stretch relation, but according to the model predictions, the effect of crystallisation is not sufficient to explain the mechanical hysteresis observed in the tensile tests. Hence, additional viscoelasticity associated with amorphous polymer chains must be included. The model was fully able to predict both the stress vs. stretch relations and the crystallinity vs. stretch relations from the experiments.

  • 16.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    A continuum mechanics framework and a constitutive model for remodelling of collagen gels and collagenous tissues2010In: Journal of the mechanics and physics of solids, ISSN 0022-5096, E-ISSN 1873-4782, Vol. 58, no 6, p. 918-933Article in journal (Refereed)
    Abstract [en]

    Collagen is a very important protein of the human body and is responsible for the structural stability of many body components. Furthermore, collagen fibre networks are able to grow and remodel themselves, which enables them to adjust to varying physiological conditions. This remodelling is accomplished by fibre-producing cells, such as fibroblasts. The ability to adjust to new physiological conditions is very important, for example in wound healing. In the present paper, a theoretical framework for modelling collagenous tissues and collagen gels is proposed. Continuum mechanics is employed to describe the kinematics of the collagen, and affine deformations of fibres are assumed. Biological soft tissues can be approximated as being hyperelastic, and the constitutive model for the collagen fabric is therefore formulated in terms of a strain energy function. This strain energy function includes a density function that describes the distribution of the collagen fibre orientation. The density function evolves according to an evolution law, where fibres tend to reorient towards the direction of maximum Cauchy stress. The remodelling of the collagen network is also assumed to include a pre-stretching of collagen fibres, accomplished by fibroblasts. The theoretical framework is applied to experiments performed on collagen gels, where gels were exposed to remodelling under both biaxial and uniaxial constraints. The proposed model was able to predict both the resulting collagen distribution and the resulting stress-strain relationships obtained for the remodelled collagen gels. The influence of the most important model parameters is demonstrated, and it appears that there is a fairly unique set of model parameters that gives an optimal fit to the experimental data.

  • 17.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    A numerical framework for material characterisation of inhomogeneous hyperelastic membranes by inverse analysis2010In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 234, no 2, p. 563-578Article in journal (Refereed)
    Abstract [en]

    An inverse method for estimating the distributions of the elastic properties of hyperelastic, inhomogeneous membranes is proposed. The material description of the membrane is based on a versatile constitutive model, in which two stiffness parameters govern the nonlinear elastic behaviour of the material. The estimation procedure includes a finite element framework. The two stiffness parameters in the constitutive law are assumed to vary continuously over the inhomogeneous membrane, and in the finite element framework the distributions of the two parameters are approximated using standard linear shape functions. Experimental results are assumed to exist in terms of nodal displacements from a test with known geometry and boundary conditions. The experimental membrane is modelled in the finite element framework, and the deformation of it is predicted. An error function, quantifying the discrepancy between the experimentally obtained deformation pattern and the numerically predicted pattern, is then minimised with respect to the nodal values of the two interpolated parameters. In a number of numerical examples, the proposed procedure is assessed by attempting to reproduce the given random reference distributions of material properties. The proposed estimation method is fully able to reproduce the reference distributions of the two material parameters with excellent accuracy.

  • 18.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    A theoretical assessment of the influence of myosin filament dispersion on smooth muscle contraction2011In: ASME 2011 Summer Bioengineering Conference, 2011, no PARTS A AND B, p. 145-146Conference paper (Refereed)
    Abstract [en]

    A new constitutive model for the biomechanical behavior of smooth muscle tissue is employed to investigate the influence of statistical dispersion in the orientation of myosin filaments. The number of activated cross-bridges between the actin and myosin filaments governs the contractile force generated by the muscle and also the contraction speed. A strain-energy function is used to describe the mechanical behavior of the smooth muscle tissue. The predictions from the constitutive model are compared to histological and isometric tensile test results for smooth muscle tissue from swine carotid artery. In order to be able to predict the active stress at different muscle lengths, a filament dispersion significantly larger than the one observed experimentally was required. Furthermore, a comparison of the predicted active stress for a case of uniaxially oriented myosin filaments and a case of filaments with a dispersion based on the experimental histological data shows that the difference in generated stress is noticeable but limited. Thus, the results suggest that myosin filament dispersion alone cannot explain the increase in active muscle stress with increasing muscle stretch. Copyright © 2011 by ASME.

  • 19.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    An 8-chain Model for Rubber-like Materials Accounting for Non-affine Chain Deformations and Topological Constraints2011In: Journal of elasticity, ISSN 0374-3535, E-ISSN 1573-2681, Vol. 102, no 2, p. 99-116Article in journal (Refereed)
    Abstract [en]

    Several industrial applications involve rubber and rubber-like materials, and it is important to be able to predict the constitutive response of these materials. In the present paper, a new constitutive model for rubber-like solids is proposed. The model is based on the 8-chain concept introduced by Arruda and Boyce (J. Mech. Phys. Solids 41, 389-412, 1993) to which two new components are added. Real polymer networks do not deform affinely, and in the proposed model this is accounted for by the inclusion of an elastic spring, acting in series with the representative polymer chain. Furthermore, real polymer chains are not completely free to move, which is modelled by imposing a topological constraint on the transverse motions of the representative polymer chain. The model contains five model parameters and these need to be determined on the basis of experimental data. Three: experimental studies from the literature were used to assess the proposed model. The model was able to reproduce experimental data performed under conditions of uniaxial tension, generalised plane deformation, and biaxial tension with an excellent accuracy. The strong predictive abilities together with the numerically efficient structure of the model make it suitable for implementation in a finite element context.

  • 20.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    An asymptotic analysis of dynamic crack growth in rubber2011In: Engineering Fracture Mechanics, ISSN 0013-7944, E-ISSN 1873-7315, Vol. 78, no 17, p. 3111-3122Article in journal (Refereed)
    Abstract [en]

    Asymptotic analyses of the mechanical fields in front of stationary and propagating cracks are important for several reasons. For example, they facilitate the understanding of the mechanical and physical state in front of crack tips, and they enable prediction of crack growth. Furthermore, efficient modelling of arbitrary crack growth by use of XFEM (extended finite element method) requires accurate knowledge of the asymptotic crack tip fields. The present study focuses on the asymptotic fields in front of a crack that propagates dynamically in rubber. Static analyses of this type of problem have been made in previous studies. In order to be able to compare the present results with these earlier studies, the constitutive model from Knowles and Sternberg (J. Elast. 3: 67-107, 1973) was adopted. It is assumed that viscoelastic stresses become negligible compared with the singular elastic stresses close to the crack tip. The present analysis shows that in materials with a significant hardening, the inertia term in the equations of motion becomes negligible in the asymptotic analysis. However, for a neoHookean type of model, inertia comes into play and causes a maximum theoretical crack speed that equals the shear wave speed.

  • 21.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    An Efficient Method for Material Characterisation of Hyperelastic Anisotropic Inhomogeneous Membranes Based on Inverse Finite-Element Analysis and an Element Partition Strategy2010In: Quarterly Journal of Mechanics and Applied Mathematics, ISSN 0033-5614, E-ISSN 1464-3855, Vol. 63, no 2, p. 201-225Article in journal (Refereed)
    Abstract [en]

    An inverse method for estimating the distributions of the nonlinear elastic properties of inhomogeneous and anisotropic membranes is investigated. The material description of the membrane is based on a versatile constitutive model, including four material parameters: two initial stiffness values pertaining to the principal directions of the material, the angle between these principal directions and a reference coordinate system and a parameter related to the level of nonlinearity of the material. The estimation procedure consists of the following three steps: (i) perform experiments on the membranous structure whose properties are to be determined, (ii) define a corresponding finite-element (FE) model and (iii) minimise an error function (with respect to the unknown parameters) that quantifies the deviation between the numerical predictions and the experimental data. For this finite deformation problem, an FE framework for membranous structures exposed to a pressure boundary loading is outlined: the principle of virtual work, its linearisation and the related spatial discretisation. To achieve a robust parameter estimation, an element partition method is employed. In numerical examples, the proposed procedure is assessed by attempting to reproduce given random reference distributions of material fields in a reference membrane. The deviations between the estimated material parameter distributions and the related reference fields are within a few percent in most cases. The standard deviation for the resulting maximum principal stress was consistently below 1%.

  • 22.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    ASSESSMENT OF THREE POSSIBLE CRITERIA FOR REMODELLING OF COLLAGEN GELS AND COLLAGENOUS TISSUES2010In: PROCEEDINGS OF THE ASME SUMMER BIOENGINEERING CONFERENCE, 2010, NEW YORK: ASME , 2010, p. 775-776Conference paper (Refereed)
  • 23.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Asymptotic mechanical fields at the tip of a mode I crack in rubber-like solids2014In: International Journal of Solids and Structures, ISSN 0020-7683, E-ISSN 1879-2146, Vol. 51, no 10, p. 1923-1930Article in journal (Refereed)
    Abstract [en]

    Asymptotic analyses of the mechanical fields in front of stationary and propagating cracks facilitate the understanding of the mechanical and physical state in front of crack tips, and they enable prediction of crack growth and failure. Furthermore, efficient modelling of arbitrary crack growth by use of XFEM (extended finite element method) requires accurate knowledge of the asymptotic crack tip fields. In the present work, we perform an asymptotic analysis of the mechanical fields in the vicinity of a propagating mode I crack in rubber. Plane deformation is assumed, and the material model is based on the Langevin function, which accounts for the finite extensibility of polymer chains. The Langevin function is approximated by a polynomial, and only the term of the highest order contributes to the asymptotic solution. The crack is predicted to adopt a wedge-like shape, i.e. the crack faces will be straight lines. The angle of the wedge and the order of the stress singularity depend on the hardening of the strain energy function. The present analysis shows that in materials with a significant hardening, the inertia term in the equations of motion becomes negligible in the asymptotic analysis. Hence, there is no upper theoretical limit to the crack speed.

  • 24.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Dynamic steady-state analysis of crack propagation in rubber-like solids using an extended finite element method2012In: Computational Mechanics, ISSN 0178-7675, E-ISSN 1432-0924, Vol. 49, no 1, p. 73-86Article in journal (Refereed)
    Abstract [en]

    In the present study, a computational framework for studying high-speed crack growth in rubber-like solids under conditions of plane stress and steady-state is proposed. Effects of inertia, viscoelasticity and finite strains are included. The main purpose of the study is to examine the contribution of viscoelastic dissipation to the total work of fracture required to propagate a crack in a rubber-like solid. The computational framework builds upon a previous work by the present author (Kroon in Int J Fract 169:49-60, 2011). The model was fully able to predict experimental results in terms of the local surface energy at the crack tip and the total energy release rate at different crack speeds. The predicted distributions of stress and dissipation around the propagating crack tip are presented. The predicted crack tip profiles also agree qualitatively with experimental findings.

  • 25.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Energy release rates in rubber during dynamic crack propagation2014In: International Journal of Solids and Structures, ISSN 0020-7683, E-ISSN 1879-2146, Vol. 51, no 25-26, p. 4419-4426Article in journal (Refereed)
    Abstract [en]

    The theoretical understanding of the fracture mechanics of rubber is not as well developed as for other engineering materials, such as metals. The present study is intended to further the understanding of the dissipative processes that take place in rubber in the vicinity of a propagating crack tip. This dissipation contributes significantly to the total fracture toughness of the rubber and is therefore of great interest from a fracture mechanics point of view. To study this, a computational framework for analysing high-speed crack growth in a biaxially stretched rubber under plane stress is therefore formulated. The main purpose is to investigate the energy release rates required for crack propagation under different modes of biaxial stretching. The results show, that inertia comes into play when the crack speed exceeds about 50 m/s. The total work of fracture by far exceeds the surface energy consumed at the very crack tip, and the difference must be attributed to dissipative damage processes in the vicinity of the crack tip. The size of this damage/dissipation zone is expected to be a few millimetres.

  • 26.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Influence of dispersion in myosin filament orientation and anisotropic filament contractions in smooth muscle2011In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 272, no 1, p. 72-82Article in journal (Refereed)
    Abstract [en]

    A new constitutive model for the biomechanical behaviour of smooth muscle tissue is proposed. The active muscle contraction is accomplished by the relative sliding between actin and myosin filaments, comprising contractile units in the smooth muscle cells. The orientation of the myosin filaments, and thereby the contractile units, are taken to exhibit a statistical dispersion around a preferred direction. The number of activated cross-bridges between the actin and myosin filaments governs the contractile force generated by the muscle and also the contraction speed. A strain-energy function is used to describe the mechanical behaviour of the smooth muscle tissue. Besides the active contractile apparatus, the mechanical model also incorporates a passive elastic part. The constitutive model was compared to histological and isometric tensile test results for smooth muscle tissue from swine carotid artery. In order to be able to predict the active stress at different muscle lengths, a filament dispersion significantly larger than the one observed experimentally was required. Furthermore, a comparison of the predicted active stress fora case of uniaxially oriented myosin filaments and a case of filaments with a dispersion based on the experimental histological data shows that the difference in generated stress is noticeable but limited. Thus, the results suggest that myosin filament dispersion alone cannot explain the increase in active muscle stress with increasing muscle stretch.

  • 27.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Modeling of Fibroblast-Controlled Strengthening and Remodeling of Uniaxially Constrained Collagen Gels2010In: Journal of Biomechanical Engineering, ISSN 0148-0731, E-ISSN 1528-8951, Vol. 132, no 11, p. 111008-Article in journal (Refereed)
    Abstract [en]

    A theoretical model for the remodeling of collagen gels is proposed. The collagen fabric is modeled as a network of collagen fibers, which in turn are composed of collagen fibrils. In the model, the strengthening of collagen fabric is accomplished by fibroblasts, which continuously recruit and attach more collagen fibrils to existing collagen fibers. The fibroblasts also accomplish a reorientation of collagen fibers. Fibroblasts are assumed to reorient collagen fibers toward the direction of maximum material stiffness. The proposed model is applied to experiments in which fibroblasts were inserted into a collagen gel. The model is able to predict the force-strain curves for the experimental collagen gels, and the final distribution of collagen fibers also agrees qualitatively with the experiments.

  • 28.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    On the correlation between continuum mechanics entities and cell activity in biological soft tissues: Assessment of three possible criteria for cell-controlled fibre reorientation in collagen gels and collagenous tissues2010In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 264, no 1, p. 66-76Article in journal (Refereed)
    Abstract [en]

    The biomechanical behaviour of biological cells is of great importance in many physiological processes. One such process is the maintenance of fibrous networks, such as collagenous tissues. The activity of the fibre-producing cells in this type of tissue is very important, and a comprehensive material description needs to incorporate the activity of the cells. In biomechanics, continuum mechanics is often employed to describe deforming solids, and modelling can be much simplified if continuum mechanics entities, such as stress and strain, can be correlated with cell activity. To investigate this, a continuum mechanics framework is employed in which remodelling of a collagen gel is modelled. The remodelling is accomplished by fibroblasts, and the activity of the fibroblasts is linked to the continuum mechanics theory. The constitutive model for the collagen fabric is formulated in terms of a strain energy function, which includes a density function describing the distribution of the collagen fibre orientation. This density function evolves according to an evolution law, where fibroblasts reorient fibres towards the direction of increasing Cauchy stress, elastic deformation, or stiffness. The theoretical framework is applied to experimental results from collagen gels, where gels have undergone remodelling under both biaxial and uniaxial constraint. The analyses indicated that criteria 1 and 2 (Cauchy stress and elastic deformations) are able to predict the collagen fibre distribution after remodelling, whereas criterion 3 (current stiffness) is not. This conclusion is, however, tentative and pertains, strictly speaking, only to fibre remodelling processes, and may not be valid for other types of cell activities.

  • 29.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Optimal length of smooth muscle assessed by a microstructurally and statistically based constitutive model2011In: Computer Methods in Biomechanics and Biomedical Engineering, ISSN 1025-5842, E-ISSN 1476-8259, Vol. 14, no 1, p. 43-52Article in journal (Refereed)
    Abstract [en]

    Smooth muscle exhibits an optimal length at which it is able to generate a maximum amount of force. In this study, the optimal length is assessed by use of a microstructurally and statistically based constitutive model for smooth muscle. The model is based on the sliding filament theory, and a modified version of Hill's mechanical model was adopted. It was conjectured, that a variation in the overlap in the actomyosin contractile units together with a statistical dispersion in the size of the dense bodies are responsible for the optimal length characteristics. The influence of contractile unit length, dense body size and dense body compliance was investigated, and the model was fully able to predict experimental data. The results indicate that the compliance of the dense bodies does not contribute significantly to the total compliance of the contractile apparatus.

  • 30.
    Kroon, Martin
    KTH, Superseded Departments, Solid Mechanics.
    Probabilistic and micromechanical modelling of cleavage fracture2003Doctoral thesis, comprehensive summary (Other scientific)
  • 31.
    Kroon, Martin
    KTH, Superseded Departments, Solid Mechanics.
    Probabilistic modelling of cleavage fracture2001Licentiate thesis, comprehensive summary (Other scientific)
  • 32.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Simulation of cerebral aneurysm growth and prediction of evolving rupture risk2011In: Modelling and Simulation in Engineering, ISSN 1687-5591, Vol. 2011, p. 289523-Article in journal (Refereed)
    Abstract [en]

    Cerebral aneurysms are local expansions of blood vessel walls in the brain blood system. The rupture of an aneurysm is a very severe event associated with a high rate of mortality. When cerebral aneurysms are detected, clinicians need to decide if operation is required. The risk of aneurysm rupture is then compared to the risks associated with the medical intervention. In the present paper, a probabilistic framework for a mechanically based rupture risk assessment of cerebral aneurysms is proposed. The method is based on the assumption that the strength of aneurysmal tissues can be described by a statistical distribution. A structural analysis of the aneurysm in question is performed, and the maximum stress experienced by the aneurysm is compared to the strength distribution. The proposed model was compared with clinical results for ruptured aneurysms in terms of rupture density and accumulated rupture risk as a function of aneurysm size. The model was able to reproduce the clinical results well. The proposed framework may potentially be used under in vivo conditions to predict the risk of rupture for diagnosed aneurysms.

  • 33.
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Steady-state crack growth in rubber-like solids2011In: International Journal of Fracture, ISSN 0376-9429, E-ISSN 1573-2673, Vol. 169, no 1, p. 49-60Article in journal (Refereed)
    Abstract [en]

    The fracture toughness of rubber-like materials depends on several factors. First there is the surface energy required to create new crack surface at the crack tip. Second, a significant amount of energy is dissipated through viscoelastic processes in the bulk material around the crack tip. Third, if the crack propagates very rapidly, inertia effects will come into play and contribute to the fracture toughness. In the present study, a computational framework for studying high-speed crack growth in rubber-like solids under conditions of steady-state is proposed. Effects of inertia, viscoelasticity and finite strains are included. The main purpose of the study is to study the contribution of viscoelastic dissipation to the total work of fracture required to propagate a crack in a rubber-like solid. The model was fully able to predict experimental results in terms of the local surface energy at the crack tip and the total energy release rate at different crack speeds. In addition, the predicted distributions of stress and dissipation around the propagating crack tip are presented.

  • 34.
    Kroon, Martin
    et al.
    KTH, Superseded Departments, Solid Mechanics.
    Faleskog, Jonas
    KTH, Superseded Departments, Solid Mechanics.
    A probabilistic model for cleavage fracture with a length scale-influence of material parameters and constraint2002In: International Journal of Fracture, ISSN 0376-9429, E-ISSN 1573-2673, Vol. 118, no 2, p. 99-118Article in journal (Refereed)
    Abstract [en]

    A probabilistic model for the cumulative probability of failure by cleavage fracture with a material related length scale is developed in this study. The model aims at describing the random nature of fracture in ferritic steels in the brittle-to-ductile transition temperature region. The model derives from use of an exponential function to describe the distribution of microstructural entities eligible to take part in the fracture initiation process, where also a dependence on effective plastic strain is incorporated. A nonlocal stress measure, calculated as the average stress in a spherical volume, drives the contribution to failure probability of an infinitesimal material volume. The radius of the spherical volume enters as the material length in this model. This length has a significant influence on failure probability predictions in geometries exposed to strong stress gradients as found ahead of cracks. The material length is associated with a fracture toughness threshold value. In a fracture application three model parameters need to be estimated based on testing; a parameter directly related to the mean fracture toughness, a parameter that primarily is related to crack-tip constraint effects and the material length parameter. The model is explored in a parametric study showing model features in concord with typical features found in toughness distributions from fracture mechanics testing in the transition region.

  • 35.
    Kroon, Martin
    et al.
    KTH, Superseded Departments, Solid Mechanics.
    Faleskog, Jonas
    KTH, Superseded Departments, Solid Mechanics.
    Influence of carbide/ferrite interface and carbide shape on cleavage fracture initiation in ferritic steels2003Report (Other academic)
  • 36.
    Kroon, Martin
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    Faleskog, Jonas
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    Influence of crack deflection into the carbide/ferrite interface on cleavage fracture initiation in ferritic steels2008In: Mechanics of materials (Print), ISSN 0167-6636, E-ISSN 1872-7743, Vol. 40, no 9, p. 695-707Article in journal (Refereed)
    Abstract [en]

    n this and a companion study (Kroon, M., Faleskog, J., 2005. Micromechanics of cleavage fracture initiation in ferritic steels by carbide cracking. J. Mech. Phys. Solids 53, 171–196), the initiation of cleavage fracture in ferritic steels is studied. The initiation is modelled explicitly in the form of a microcrack, which nucleates in a brittle carbide and propagates into the surrounding ferrite. The carbide is modelled as an elastic cylinder and the ferrite as an elastic viscoplastic material. The crack growth is modelled using a cohesive surface, in which the tractions are governed by a modified exponential cohesive law. The advancing microcrack, which has nucleated in the carbide, may either continue into the ferrite or deflect into the interface between the carbide and the ferrite. Special attention is given to the influence of the mode mixity factor β, which is defined as the ratio between the shear and tensile strength of the interface between the carbide and the ferrite. Crack growth in the interface occurs in shear mode and is driven by a fibre loading mechanism. For mode mixity values β⩽0.2, the crack deflects into the interface. The results indicate that crack growth in the interface can have a profound influence on the macroscopic fracture toughness of ferritic steels.

  • 37.
    Kroon, Martin
    et al.
    KTH, Superseded Departments, Solid Mechanics.
    Faleskog, Jonas
    KTH, Superseded Departments, Solid Mechanics.
    Influence of plastic rate sensitivity on cleavage fracture initiation in ferritic steels2003Report (Other academic)
  • 38.
    Kroon, Martin
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    Faleskog, Jonas
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Micrornechanics of cleavage fracture initiation in ferritic steels by carbide cracking2005In: Journal of the mechanics and physics of solids, ISSN 0022-5096, E-ISSN 1873-4782, Vol. 53, no 1, p. 171-196Article in journal (Refereed)
    Abstract [en]

    Cleavage fracture in ferritic steels is often initiated in brittle carbides randomly distributed in the material. The carbides break as a result of a fibre loading mechanism in which the stress levels in the carbides are raised, as the surrounding ferrite undergoes plastic deformation. The conditions in the vicinity of the nucleated micro-crack will then determine whether the crack will penetrate or be arrested by the ferrite. The ferrite is able to arrest nucleated cracks through the presence of mobile dislocations, which blunt and shield the microcrack and thus lowers the stresses at the crack tip. Hence, the macroscopic toughness of the material directly depends on the ability of the ferrite to arrest nucleated micro-cracks and in turn on the plastic rate sensitivity of the ferrite. The initiation of cleavage fracture is here modelled explicitly in the form of a micro-crack, which nucleates in a brittle carbide and propagates into the surrounding ferrite. The carbide is modelled as an elastic cylinder or in a few cases an elastic sphere and the ferrite as an elastic viscoplastic material. The crack growth is modelled using a cohesive surface, where the tractions are governed by a modified exponential cohesive law. It is shown that the critical stress, required to propagate a microcrack from a broken carbide, increases with decreasing plastic rate sensitivity of the ferrite. The results also show that a low stress triaxiality and a high aspect ratio of the carbide promote the initiation of cleavage fracture from a broken carbide.

  • 39.
    Kroon, Martin
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Faleskog, Jonas
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Numerical implementation of a J(2)- and J(3)-dependent plasticity model based on a spectral decomposition of the stress deviator2013In: Computational Mechanics, ISSN 0178-7675, E-ISSN 1432-0924, Vol. 52, no 5, p. 1059-1070Article in journal (Refereed)
    Abstract [en]

    A new plasticity model with a yield criterion that depends on the second and third invariants of the stress deviator is proposed. The model is intended to bridge the gap between von Mises' and Tresca's yield criteria. An associative flow rule is employed. The proposed model contains one new non-dimensional key material parameter, that quantifies the relative difference in yield strength between uniaxial tension and pure shear. The yield surface is smooth and convex. Material strain hardening can be ascertained by a standard uniaxial tensile test, whereas the new material parameter can be determined by a test in pure shear. A fully implicit backward Euler method is developed and presented for the integration of stresses with a tangent operator consistent with the stress updating scheme. The stress updating method utilizes a spectral decomposition of the deviatoric stress tensor, which leads to a stable and robust updating scheme for a yield surface that exhibits strong and rapidly changing curvature in the synoptic plane. The proposed constitutive theory is implemented in a finite element program, and the influence of the new material parameter is demonstrated in two numerical examples.

  • 40.
    Kroon, Martin
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    Faleskog, Jonas
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Öberg, Hans
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    A probabilistic model for cleavage fracture with a length scale - Parameter estimation and predictions of growing crack experiments2008In: Engineering Fracture Mechanics, ISSN 0013-7944, E-ISSN 1873-7315, Vol. 75, no 8, p. 2398-2417Article in journal (Refereed)
    Abstract [en]

    A probabilistic model for the cumulative probability of failure by cleavage fracture was applied to experimental results where cleavage fracture was preceded by ductile crack growth. The model, introduced by Kroon and Faleskog [Kroon M, Faleskog J. A probabilistic model for cleavage fracture with a length scale - influence of material parameters and constraint. Int J Fract 2002; 118:99-118], includes a non-local stress with an associated material related length scale, and it also includes a strain measure to account for the number of nucleated cleavage initiation sites. The experiments were performed on single edge cracked bend test specimens with three different crack lengths at the temperature 85 degrees C, which is in the upper transition region for the steel in question. The ductile rupture process is modelled using tile cell model for nonlinear fracture mechanics. The original cleavage fracture model had to be modified in order to account for the substantial number of cleavage initiators being consumed by the ductile process. With this modification, the model was able to accurately capture the experimental failure probability distribution.

  • 41.
    Kroon, Martin
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    Holzapfel, Gerhard A.
    A model for saccular cerebral aneurysm growth by collagen fibre remodelling2007In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 247, no 4, p. 775-787Article in journal (Refereed)
    Abstract [en]

    The first structural model for saccular cerebral aneurysm growth is proposed. It is assumed that the development of the aneurysm is accompanied by a loss of the media, and that only collagen fibres provide load-bearing capacity to the aneurysm wall. The aneurysm is modelled as an axisymmetric multi-layered membrane, exposed to an inflation pressure. Each layer is characterized by an orientation angle, which changes between different layers. The collagen fibres and fibroblasts within a specific layer are perfectly aligned. The growth and the morphological changes of the aneurysm are accomplished by the turnover of collagen. Fibroblasts are responsible for collagen production, and the related deformations are assumed to govern the collagen production rate. There are four key parameters in the model: a normalized pressure, the number of layers in the wall, an exponent in the collagen mass production rate law, and the pre-stretch under which the collagen is deposited. The influence of the model parameters on the aneurysmal response is investigated, and a stability analysis is performed. The model is able to predict clinical observations and mechanical test results, for example, in terms of predicted aneurysm size, shape, wall stress and wall thickness.

  • 42.
    Kroon, Martin
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    Holzapfel, Gerhard A.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    A new constitutive model for multi-layered collagenous tissues2008In: Journal of Biomechanics, ISSN 0021-9290, E-ISSN 1873-2380, Vol. 41, no 12, p. 2766-2771Article in journal (Refereed)
    Abstract [en]

    Collagenous tissues Such as the aneurysmal wall or the aorta are multi-layered structures with the mean fibre alignments distinguishing one layer from another. A constitutive representation of the multiple collagen layers is not yet developed, and hence the aim of the present study. The proposed model is based on the constitutive theory of finite elasticity and is characterized by an anisotropic strain-energy function which takes the material structure into account. The passive tissue behaviour is modelled and the related mechanical response is assumed to be dominated by elastin and collagen. While elastin is modelled by the neo-Hookean material the constitutive response of collagen is assumed to be transversely isotropic for each individual layer and based on an exponential function. The proposed constitutive function is polyconvex which ensures material stability. The model has five independent material parameters, each of which has a clear physical interpretation: the initial stiffnesses of the collagen fabric in the two principal directions, the shear modulus pertaining to the non-collagenous matrix material, a parameter describing the level of nonlinearity of the collagen fabric, and the angle between the principal directions of the collagen fabric and the reference coordinate system. An extension-inflation test of the adventitia of a human femoral artery is simulated by means of the finite element method and an error function is minimized by adjusting the material parameters yielding a good agreement between the model and the experimental data.

  • 43.
    Kroon, Martin
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    Holzapfel, Gerhard A.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    A theoretical model for fibroblast-control led growth of saccular cerebral aneurysms2009In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 257, no 1, p. 73-83Article in journal (Refereed)
    Abstract [en]

    A new theoretical model for the growth of saccular cerebral aneurysms is proposed by extending the recent constitutive framework of Kroon and Holzapfel [2007a. A model for saccular cerebral aneurysm growth by collagen fibre remodelling. J. Theor. Biol. 247, 775-787]. The continuous turnover of collagen is taken to be the driving mechanism in aneurysmal growth. The collagen production rate depends on the magnitude of the cyclic deformation of fibroblasts, caused by the pulsating blood pressure during the cardiac cycle. The volume density of fibroblasts in the aneurysmal tissue is taken to be constant throughout the growth process. The growth model is assessed by considering the inflation of an axisymmetric membranous piece of aneurysmal tissue, with material characteristics representative of a cerebral aneurysm. The diastolic and systolic states of the aneurysm are computed, together with its load-free state. It turns out that the value of collagen pre-stretch, that determines growth speed and stability of the aneurysm, is of pivotal importance. The model is able to predict aneurysms with typical berry-like shapes observed clinically, and the predicted wall stresses correlate well with the experimentally obtained ultimate stresses of this type of tissue. The model predicts that aneurysms should fail when reaching a size of about 1.2-3.6 mm, which is smaller than what has been clinically observed. With some refinements, the model may, however, be used to predict future growth of diagnosed aneurysms.

  • 44.
    Kroon, Martin
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Holzapfel, Gerhard A.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    A theoretical model for saccular cerebral aneurysm growth: Deformation and stress-analysis2007In: Proceedings of the ASME Summer Bioengineering Conference 2007, 2007, p. 255-256Conference paper (Refereed)
  • 45.
    Kroon, Martin
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Holzapfel, Gerhard A.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Elastic properties of anisotropic vascular membranes examined by inverse analysis2009In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 198, no 45-46, p. 3622-3632Article in journal (Refereed)
    Abstract [en]

    An inverse method for estimating the distributions of the nonlinear elastic properties of inhomogeneous and anisotropic vascular membranes such as cerebral aneurysms is proposed. The material description of the membrane is based on a versatile structural model able to represent multiple collagen layers and the passive response of the vascular wall. Each individual layer is assumed to behave transversely isotropic following exponential stiffening with increasing loading. The model includes four parameters to be explainable physically: two initial stiffnesses of the collagen fabric, a parameter related to the nonlinearity of the collagen fabric, angle between the principal directions of the collagen fabric and a reference coordinate system. For this finite deformation problem a finite element framework for membranous structures considering pressure boundary loading is outlined. i.e. the principle of virtual work, its linearisation and the related spatial discretisation. The estimation procedure consists of the following three steps: (i) in vivo or in vitro approaches record the mechanical responses of membranous structures whose properties are to be determined; (ii) define a corresponding finite element model; (iii) minimise an error function (regarding the unknown parameters) that quantifies the deviation of the numerical prediction from the recorded data. To achieve a robust parameter estimation, an element partition method is employed. The outcome of the procedure is affected by the number of nodes defined on the membrane surface and the number of load steps. In a numerical example, the proposed procedure is assessed by reestablishing given reference distributions in a reference membrane. The deviations of the estimated material parameter distributions from the related reference fields are within just a few percent. In most of the investigated cases the standard deviation for the resulting maximum principal stress was even below 1%, which is accurate enough for rupture risk assessment of vascular membranes.

  • 46.
    Kroon, Martin
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    Holzapfel, Gerhard A.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    Estimation of the distributions of anisotropic, elastic properties and wall stresses of saccular cerebral aneurysms by inverse analysis2008In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 464, no 2092, p. 807-825Article in journal (Refereed)
    Abstract [en]

    A new method is proposed for estimating the elastic properties of the inhomogeneous and anisotropic structure of saccular cerebral aneurysms by inverse analysis. The aneurysm is modelled as a membrane and the constitutive response of each individual layer of the passive tissue is characterized by a transversely isotropic strain energy function of exponential type. The collagen fibres in the aneurysm wall are assumed to govern the mechanical response. Four parameters characterize the constitutive behaviour of the tissue: two initial stiffnesses of the collagen fabric in the two in-plane principal directions, one parameter describing the degree of nonlinearity that the collagen fibres exhibit and the other structural parameter, i.e. the angle which defines the orientation of the collagen fibres. The parameter describing the fibre nonlinearity is assumed to be constant, while all others are assumed to vary continuously over the aneurysm surface. Two model aneurysms, with the same initial geometry, boundary and loading conditions, constitutive behaviour and finite-element discretization, are defined: a 'reference model' with known distributions of material and structural properties and an 'estimation model' whose properties are to be estimated. An error function is defined quantifying the deviations between the deformations from the reference and the estimation models. The error function is minimized with respect to the unknown parameters in the estimation model, and in this way the reference parameter distributions are re-established. In order to achieve a robust parameter estimation, a novel element partition method is employed. The accordance between the estimated and the reference distributions is satisfactory. The deviations of the maximum stress distributions between the two models are below 1%. Consequently, the wall stresses in the cerebral aneurysm estimated by inverse analysis are accurate enough to facilitate the assessment of the risk of aneurysm rupture.

  • 47.
    Kroon, Martin
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    Holzapfel, Gerhard A.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Biomechanics.
    Modeling of saccular aneurysm growth in a human middle cerebral artery2008In: Journal of Biomechanical Engineering, ISSN 0148-0731, E-ISSN 1528-8951, Vol. 130, no 5Article in journal (Refereed)
    Abstract [en]

    Saccular aneurysm growth in a human middle cerebral artery is modeled. The aneurysm growth model was presented in a companion paper by Kroon and Holzapfel ("A Model for Saccular Cerebral Aneurysm Growth by Collagen Fibre Remodelling," J. Theor. Biol., in press) and was assessed there for axisymmetric growth. The aneurysm growth model is now evaluated for a more realistic setting. The middle cerebral artery is modeled as a two-layered cylinder, where the layers correspond to the media and the adventitia. An instant loss of the media in a region of the artery wall initiates the growth of the saccular aneurysm. The aneurysm wall is assumed to be a development of the adventitia of the original healthy artery, and collagen is assumed to be the only load-bearing constituent in the adventitia and in the aneurysm wall. The collagen is organized in a number of distinct layers where fibers in a specific layer are perfectly aligned in a certain fiber direction. The production of new collagen is taken to depend on the stretching of the aneurysm wall, and the continuous remodeling of the collagen fibers is responsible for the aneurysm growth. The general behavior of the growth model is investigated and also the influence of the structural organization of the collagen fabric. The analysis underlines the fact that the material behavior of aneurysmal tissue cannot be expected to be isotropic. The model predictions agree well with clinical and experimental results, for example, in terms of aneurysm size and shape, wall stress levels, and wall thickness.

  • 48.
    Kroon, Martin
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Holzapfel, Gerhard A.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Modelling of thin anisotropic collagen-dominated soft biological tissue2008In: ADVANCES IN HETEROGENEOUS MATERIAL MECHANICS 2008 / [ed] Fan, JH; Chen, HB, LANCASTER: DESTECH PUBLICATIONS, INC , 2008, p. 614-615Conference paper (Refereed)
    Abstract [en]

    A new constitutive model for thin, inhomogeneous, anisotropic, soft biological tissues is proposed. The thin tissue is modelled as a membrane, and the constitutive behaviour is characterized by a strain-energy function, which takes the structural features of soft biological tissue into account. It is assumed that the mechanical response of the tissue is dominated by its fibrous components, elastin and collagen. The proposed model is partly based on constitutive laws proposed by Holzapfel et al. (2000) and Kroon and Holzapfel (2007). The strain-energy of collagen fibres is modelled by use of a Fung-type strain-energy function, whereas the influence of the elastin fabric and the matrix material is modelled by use of an incompressible neo-Hookean model. In total the model has five material parameters: mu (the effective shear modulus pertaining to the elastin fabric and the matrix material), E-1 and E-2 (the initial stiffness of the collagen fabric in the two principal directions, respectively), a (a parameter describing the level of nonlinearity of the collagen fibres), and beta (the angle between the principal directions in the material and the reference coordinate system). A major advantage with the proposed model is, that the model parameters have clear physical interpretations. In addition, the total strain-energy function is polyconvex, which ensures material stability (Schroder and Neff, 2003). The model is fitted to results from inflation/extension tests on an adventitia tube from a femoral artery (Schulze-Bauer et al., 2002). Figure 1 shows a comparison between model predictions and experimental results in terms of principal stretches (lambda(1) and lambda(c)) for different internal pressure loads p. The agreement between model predictions and experiments is very good.

  • 49.
    Murtada, Sae-Il
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Holzapfel, Gerhard
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    A calcium-driven mechanochemical model for prediction of force generation in smooth muscle2010In: Biomechanics and Modeling in Mechanobiology, ISSN 1617-7959, E-ISSN 1617-7940, Vol. 9, no 6, p. 749-762Article in journal (Refereed)
    Abstract [en]

    A new model for the mechanochemical response of smooth muscle is presented. The focus is on the response of the actin-myosin complex and on the related generation of force (or stress). The chemical (kinetic) model describes the cross-bridge interactions with the thin filament in which the calcium-dependent myosin phosphorylation is the only regulatory mechanism. The new mechanical model is based on Hill's three-component model and it includes one internal state variable that describes the contraction/relaxation of the contractile units. It is characterized by a strain-energy function and an evolution law incorporating only a few material parameters with clear physical meaning. The proposed model satisfies the second law of thermodynamics. The results of the combined coupled model are broadly consistent with isometric and isotonic experiments on smooth muscle tissue. The simulations suggest that the matrix in which the actin-myosin complex is embedded does have a viscous property. It is straightforward for implementation into a finite element program in order to solve more complex boundary-value problems such as the control of short-term changes in lumen diameter of arteries due to mechanochemical signals.

  • 50.
    Murtada, Sae-Il
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Kroon, Martin
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Holzapfel, Gerhard
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Modeling the dispersion effects of contractile fibers in smooth muscles2010In: Journal of the mechanics and physics of solids, ISSN 0022-5096, E-ISSN 1873-4782, Vol. 58, no 12, p. 2065-2082Article in journal (Refereed)
    Abstract [en]

    Micro-structurally based models for smooth muscle contraction are crucial for a better understanding of pathological conditions such as atherosclerosis, incontinence and asthma. It is meaningful that models consider the underlying mechanical structure and the biochemical activation. Hence, a simple mechanochemical model is proposed that includes the dispersion of the orientation of smooth muscle myofilaments and that is capable to capture available experimental data on smooth muscle contraction. This allows a refined study of the effects of myofilament dispersion on the smooth muscle contraction. A classical biochemical model is used to describe the cross-bridge interactions with the thin filament in smooth muscles in which calcium-dependent myosin phosphorylation is the only regulatory mechanism. A novel mechanical model considers the dispersion of the contractile fiber orientations in smooth muscle cells by means of a strain-energy function in terms of one dispersion parameter. All model parameters have a biophysical meaning and may be estimated through comparisons with experimental data. The contraction of the middle layer of a carotid artery is studied numerically. Using a tube the relationships between the internal pressure and the stretches are investigated as functions of the dispersion parameter, which implies a strong influence of the orientation of smooth muscle myofilaments on the contraction response. It is straightforward to implement this model in a finite element code to better analyze more complex boundary-value problems.

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